(07/08/2010, 11:31 PM)tommy1729 Wrote: for parabolic iteration i use the carleman matrix method.

What you mean by carleman method?
We have Andrews intuitive Abel function which uses the Carleman matrix as well as Gottfrieds matrix power method.

Also I dont know how you do that if you develop at a fixed point, or dont you? Matrix power method at a fixed point is equal to regular/parabolic iteration and yields a series expansion that has convergence radius 0.

(07/08/2010, 11:31 PM)tommy1729 Wrote: .....in fact i would like to see a limit like superfunction type of solution to parabolic iteration , something similar to koenigs non-parabolic iteration solution.

does there exist a method for all fixpoints ( parabolic or non-parabolic ) in terms of a limit , not using carleman ?

Actually I dont know exactly the proof, but in most cases the convergence radius is 0 for parabolic iteration, particularly for the iteration of e^x-1 which is in turn equivalent to iteration of e^(x/e).

For parabolic Abel function there is a very old formula by Lévy (2.20 in the overview paper), which is:

which however is not usable for numeric calculation as it is too slow.
A formula given by Ecalle (2.22 in the overview paper) is much more usable and works for both cases hyperbolic and parabolic. It is
where is a sum of some negative powers (none in the hyperbolic case) and a logarithm for example for e^x-1 we get:.

Another formula (2.29 in the overview paper) that kinda combines hyperbolic and parabolic is:, ,

where is the derivative at the fixed point 0, which is 1 in the parabolic case and you take the limit of lambda->1. I am in a hurry a bit. So perhaps more detailed later.

Well, there is no more much to add, if you take the limit of the above right side:
you get if I not err, which is then the parabolic Levý formula.

If you could invert for then would be another formula for the hyperbolic Abel function.
If you can't (numerically/symbolically whatever) invert then you still have a different formula for the hyperbolic (and Lévy's formla for ) superfunction/iteration: